Radon integrals. An abstract approach to integration and Riesz representation through function cones
A \(\nu:\overline\mathbb{R}^ X\to\mathbb{R}\) is called an upper functional (UF) if \(T:=\{f\in\tilde\mathbb{R}^ X:\nu(f)<\infty\}\) is a lattice cone on which \(\nu\) is positive-homogeneous with \(\nu(s+t)\leq\nu(s\wedge t)+\nu(s\lor t)\leq\nu(s)+\nu(t)\), satisfying \(\nu(f)=\inf\{\nu(t):f\leq t\in T\}\) for \(f\in\overline\mathbb{R}^ X\), with \(\tilde R:=(-\infty,\infty]\), \(X\) a set \(\neq\emptyset\). If \(\mu:S\to\tilde\mathbb{R}\) is positive-homogeneous and additive (=: linear) on a min-closed cone \(S\) from \(\tilde\mathbb{R}^ X\), then \(\mu^*(f):=\inf\{\mu(s):f\leq s\in S\}\) defines an UF. \(f\in\overline\mathbb{R}^ X\) is called \(\nu\)-integrable if \(\nu_ *(f)=\nu(f)\in\mathbb{R}\), with \(\nu_ *(f):=-\nu(-f)\); if \(\nu=\mu^*\), \(f\) is \(\mu^*\)-integrable iff to each \(\varepsilon>0\) there are \(s,t\in S\) with \(-s\leq f\leq t\) and \(\mu(s+t)<\varepsilon\) (abstract proper Riemann integral). For the set \(J(\nu)\) of \(\nu\)-integrable functions the usual linear and lattice theory is obtained, \(J(\nu)\) is closed in \(\overline\mathbb{R}^ X\) with respect to \(d(f,g):=\nu(| f-g|)\). (Though not directly comparable, this \(\nu\)-integral should be subsumed by the Aumann-Schäfke integration with integral metrics [\textit{G. Aumann}, Arch. Math. 3, 441-450 (1952; Zbl 0048.037), \textit{D. Hoffmann} and \textit{F.-W. Schäfke}, ``Integrale.'' Bibl. Inst. Mannheim (1992), \textit{K. Bichteler}, ``Integration theory'' (with special attention to vector measures) (1973; Zbl 0249.28002)].) \(J(\nu)\) is extended to \(J^ 0(\nu):=J(\nu^ 0)\) containing improper absolutely \(\nu\)-integrable functions, with UF \(\nu^ 0(f):=\inf_{u\in J_ -}\sup_{v\in J_ - }\nu((f\wedge(-v))\lor u)\), \(J_ -:=J(\nu)\cap(-\infty,0]^ X\) (which corresponds to Schäfke's local integral norm [\textit{F.-W. Schäfke}, J. Reine Angew. Math. 289, 118-134 (1977; Zbl 0337.28011)]; for the case of finitely additive set functions such an integral has been introduced by the reviewer [Rend. Sem. Mat. Fis. Milano 43(1973), 167-176 (1974; Zbl 0322.28010), esp. p. 172], extending the Dunford-Schwartz integral; for \(\nu=\mu^*\) with linear \(\mu\geq 0\) on a vector lattice this \(J^ 0\) has been introduced by \textit{P. Muñoz Rivas} and \textit{M. Díaz Carillo} [Acta Univ. Carol., Math. Phys. 31, 25-27 (1990)].) For UF \(\nu\), \(\nu\)-measurability in the sense of Stone is discussed, e.g. \(f\in J(\nu)\) iff \(f\) is \(\nu\)-measurable and \(\nu(| f|)<\infty\) (also in \textit{F. W. Schäfke}, loc. cit.). If the UF \(\nu\) satisfies Daniell's condition, \(\nu(\sup_ nf_ n)=\sup_ n\nu(f_ n)\) for \(f_ n\leq f_{n+1}\in\overline\mathbb{R}^ X\) (=upper integral, strong integral norm), the usual convergence theorem inclusive completeness are obtained, with \(\nu\) also \(\nu^ 0\) is an upper integral, but \(J^ 0=J\bmod\nu^ 0\)-nulsets; the \(\nu\)- resp. \(\nu^ 0\)-measurable sets form a \(\delta\)-ring, a \(\sigma\)-algebra if 1 is \(\nu\)-measurable. All this is discussed especially for \(\nu=\mu^*\) with \(\mu| S\) as above, furthermore integration with respect to \(\nu_ 1+\nu_ 2\), \(\sum^ \infty_ 1\nu_ n\), \(m\cdot\nu\) \((m\) weight function) and \(p(\nu)\), \(p:X\to Y\), is characterized. For a given linear \(\tau:T\to\tilde\mathbb{R}\) conditions for representation by another linear \(\mu| S\) (given or constructed) are obtained, i.e., \(\tau=(\mu^ 0)_ *=\mu^ 0\) on \(T\), \(\mu^ 0:=(\mu^*)^ 0\), with cones \(T\), \(S\). If \(X\) is any Hausdorff space, the cone \(S(X):=\) all lower semicontinuous \(\tilde\mathbb{R}\)-valued functions \(\geq 0\) outside some compact set; a \(\mu:S(X)\to\tilde\mathbb{R}\) is called a Radon integral if it is linear and regular, i.e., \(\mu=\mu_ *\) on \(S(X)\), \(\mu_ *(f):=-\mu^*(-f)\). Every Radon integral is a Bourbaki integral, i.e., Daniell's condition holds for increasing nets from \(S(X)\) instead of sequences; example: Lebesgue integral restricted to \(S(\mathbb{R}^ n)\). For locally compact \(X\), any non-negative linear \(\mu\) on the continuous functions with compact support, resp. vanishing at \(\infty\), is a Bourbaki integral, via extension corresponding bijectively with the Radon integrals; similar results hold for completely regular \(X\). For metrizable topological \(X\) and Radon integrals, the theorems of Egoroff and Lusing hold. If the monotone linear \(\mu:S\to\tilde\mathbb{R}\) is regular (always true for vector spaces \(S)\) and satisfies Daniell's condition on \(S\), then the usual extension \(\mu^ \sigma\) to monotone limits from \(S\) gives UFs \(\nu=\mu^{\sigma *}\), \(\nu^ 0\) satisfying Daniell's condition; the usual Daniell-\(\mu\)-integrable functions are thus obtained by the above abstract Riemann closure \(J(\nu)\) from the cone of monotone limits from \(S\). With a suitable extension of Daniell's condition, also non-regular \(\mu\) can be treated. If \({\mathcal K}\) is a lattice of sets from \(X\), an \(f:X\to\overline\mathbb{R}\) is called \({\mathcal K}\)-measurable if to each \(K\in{\mathcal K}\) and real \(\gamma,\delta\) with \(\gamma<\delta\) there is \(L\in{\mathcal K}\) with \(\{f\leq\gamma\}\cap K\subset L\subset\{f\leq\delta\}\cap K\); under various assumptions, properties of the set of \({\mathcal K}\)-measurable functions and step functions \(E({\mathcal K})\) \((f(X)\) finite, \(f\;\) \({\mathcal K}\)-measurable and \(\geq 0\) outside some \(K\in{\mathcal K})\) are deduced, especially for \({\mathcal K}(S):=\{\{s\leq -1\}:s\in S\}\) for Stonean \(S\), or \({\mathcal K}(X):=\{K\) compact\} for topological \(X\). If \(m:{\mathcal K}\to[0,\infty]\) is given, conditions on \(m\) \((m\) `content') are formulated such that it can be extended to a monotone linear \(\mu:E({\mathcal K})\to\tilde\mathbb{R}\), resp. monotone additive on some given \({\mathcal M}\subset\) algebra generated by \({\mathcal K}\). By Loomis' countable spectrum lemma, \(f\) is \(\mu^*\)- measurable iff it is \({\mathcal R}^ X\)-measurable, \({\mathcal R}^ X:=\mu^*\)-integrable sets. `\({\mathcal G}\)-regularity' of \(m|{\mathcal K}\) extends to \(\mu\) and \(\mu^*\) if \({\mathcal K}\), \({\mathcal G}\) are `compatible' (imitating compact and open sets). If a Radon measure is a \({\mathcal G}(X)\)- regular content on \({\mathcal K}(X)\), \({\mathcal G}(X):=\)\{open sets\}, then for Hausdorff \(X\) there is a canonical bijection between Radon integrals and Radon measures. \(m|{\mathcal K}\) is \(\sigma\)-additive (suitably defined) iff the above extension \(\mu\) satisfies Daniell's condition, similarly for Bourbaki's condition. If \(T\) is a Stonean lattice cone \(\subset\overline\mathbb{R}^ X\), \(\tau| T\) monotone linear and \({\mathcal K}\) and \({\mathcal G}\) are compatible lattices of sets from \(X\), sufficient conditions for the existence of a (regular) content \(m|{\mathcal K}\) are given such that \(m\) represents \(\tau\); with this general representation theorem, UFs representable by contents can be described, one has a bijection between \(\tau\) and \(m\) on \({\mathcal K}(T)\); the representation theorems of Alexandroff (dual of the bounded continuous functions), Markov \((X\) normal), Riesz (\(X\) locally compact) and Bauer follow (compare also with the reviewer [Rend. Sem. Mat. Fis. Milano 45(1975), 107-168 (1976; Zbl 0369.28004), Linear Operators Approx. II, Proc. Conf. Oberwolfach 1974, ISNM 25, 113-126 (1974; Zbl 0304.28007)]. Analogously the Daniell-Stone representation theorem is obtained. An Appendix contains historical remarks and a somewhat selected list of earlier results in this direction. Fubini theorems are not treated. In the spirit of the series `progress in mathematics' the style of the book is mostly that of a paper. Many (occasionally ambiguous) notations and theorems without assumptions (they are usually collected somewhere before) make the reading somewhat difficult.
scientific article; zbMATH DE number 52672